5. Goals
• Establish a clear connection between operations with
whole numbers and fractions
• Engage in contextual problems
• Reason/sense-making of operations involving fractions
• Reflect on “your” thinking as you do the problems
• Reflect on important language
6. Mathematics Teaching Practices
• Establish mathematics goals to focus learning.
• Implement tasks that promote reasoning and
problem solving.
• Use and connect mathematical representations.
• Facilitate meaningful mathematical discourse.
• Pose purposeful questions.
• Build procedural fluency from conceptual
understanding.
• Support productive struggle in learning mathematics.
• Elicit and use evidence of student thinking.
7. Volume Lowering Strategies
• Strategy #1: Notation/Comparison
• Strategy #2: Context
• Strategy #3: Connect to Whole Number Problems
8. Tensions
• “Their” models vs. “Our” models
• Concrete vs. Abstract
• Notation and Language
• Fractions as part-whole, scalar, operator, ratio
10. How do we “interpret”
2
3
?
• “2 over 3”
• ”2 thirds”
• 2 “copies” or “groups of”
1
3
• “multiply by 2 and divided by 3”
• ”2 hits in 3 total at-bats”
• ”2 parts out of 3 total parts”
11. How do we “write” it?
• 2 thirds
• 2 *
1
3
• 2
1
3
•
2
3
14. Reasoning Strategies
• Whole Number Reasoning-–”they are equal”
(especially when comparing
5
6
and
7
8
)
• Gap Reasoning—”one-third is 2 from the
whole and one-sixth is 5 from the whole so
one-third is greater”
16. Strategy #2 (Context)
• Drew ate 1 sixth of a whole Twizzler. Brad ate 1 third
of a whole Twizzler. Who ate more?
• Group A is going to equally share 1 Twizzler with 6
people. Group B is going to equally share 1 Twizzler
with 3 people. The people in which group will get
more get more Twizzler? How much will each a
person get in each group?
Turning the Volume Way Down!
20. Addition
• Jenny has 4 whole apples and her friend Sue
has 9 whole apples. Between the two of them
how many total whole apples do they have?
• Jenny has
1
3
of a whole pound of cheese and
her friend Sally has
3
4
of a whole pound of
cheese. Between the two of them how much
cheese do they have?
21. Addition with Whole Numbers
4 groups of 1 apple + 9 groups of 1 apple = 13 groups of 1 apple
4 whole apples + 9 whole apples = 13 whole apples
27. Thinking about “Division”
• Lily has 12 Starbursts. She is going to give them
away to her 3 friends. How many Starbursts will
each friend receive?
• Gwen has 12 Starbursts. She is going to give 3
Starbursts to as many friends as she can. How many
friend will receive 3 Starbursts?
What are the similarities/differences in the structure
of the two problems?
28. Which is “louder”?
5 ÷
1
3
OR
There are 5 whole Twizzlers and each person is
going to receive
1
3
of a whole Twizzler? Using all of
the Twizzlers, how many people can you give
1
3
of a
whole Twizzler?
29. Strategy #2: Context
There are 5 whole Twizzlers and each person is going to
receive
1
3
of a whole Twizzler? Using all of the Twizzlers,
how many people can you give
1
3
of a whole Twizzler?
30.
31.
32.
33. Problem Prompts
• Using only pictorial representations
determine the answer to each problem.
• Create a numerical expression that models
your pictorial representation.
34. Problem 1
You have 8 peanut butter and jelly sandwiches
and each student will eat
2
3
of a whole peanut
butter and jelly sandwich. How many students
can you feed?
35. Problem 2
You have 24 peanut butter and jelly sandwiches
and each student will eat 2 peanut butter and
jelly sandwiches. How many students can you
feed?
36.
37. Making the Connection
𝟖 ÷
𝟐
𝟑
• Question: How many groups
of 2 thirds can I make from
24 thirds?
• Answer: 12 groups of 2
thirds of a whole.
𝟐𝟒 ÷ 𝟐
• Question: How many groups
of 2 whole can I make from
24 whole?
• Answer: 12 groups of 2
whole.
39. Thinking about Multiplication
Lily is making gift boxes of cookies. Each gift box
will hold 5 cookies. If she wants to make 4 gift
boxes how many cookies will she need to make?
4 * 5 = 20
(multiplier) * (multiplicative unit) = product
40. Different Scenarios
• Scenario #1: Multiplier is whole number;
Multiplicative Unit is a fraction
• Scenario #2: Multiplier is a fraction; Multiplicative
Unit is a whole number
• Scenario #3: Multiplier and Multiplicative Unit are
both fractions
41. Problem 3
Each batch of cookies requires
2
3
of a whole cup
of sugar. How much sugar will be needed to
make 5 batches of cookies?
42. Problem 4
Each batch of cookies requires 2 whole cups of
sugar. How much sugar will be needed to make 5
batches of cookies?
44. Making the Connection
𝟓 ∗
𝟐
𝟑
• Question: How much is 5
groups of 2 thirds?
• Answer: 10 thirds
5*2
• Question: How much is 5
groups of 2 whole?
• Answer: 10 whole
5 is the multiplier,
2
3
and 2 are the multiplicative unit.
46. Problem 5 (The “Twist”)
Joann has 15 cups of sugar. She wants to give
2
3
of what she has to her friend Ann so Ann can
make some cookies. How much sugar will Ann
receive?
47. 2
3
* 15 (Each box represents 1 whole cup of sugar)
2
3
is the multiplier and 15 is multiplicative unit
2
3
as the multiplier implies an action on the multiplicative unit—
”partition multiplicative unit into 3 equal groups and take 2 of those
groups”
2
3
* 15 = 2 *
1
3
∗ 15 = 2 * 5 = 10
48. Moment of Reflection
Think about how the fraction
2
3
is treated in each
problem.
• Each batch of cookies requires
2
3
of a whole cup of
sugar. How much sugar will be needed to make 5
batches of cookies?
• Joann has 15 cups of sugar. She wants to give
2
3
of
what she has to her friend Ann so Ann can make
some cookies. How much sugar will Ann receive?
49. “Tame Way”
Joann has
15
16
of a whole cup of sugar. She wants
to give
2
3
of what she has to her friend Ann so
Ann can make some cookies. How much sugar
will Ann receive?
How is this problem similar/different in
structure to the problem you just solved?
50. 2
3
* 15 sixteenth (Each box represents 1 sixteenth of whole cup of sugar)
2
3
is the multiplier and 15 sixteenths, 15
1
16
, is multiplicative unit
2
3
as the multiplier implies an action on the multiplicative unit—”partition
multiplicative unit into 3 equal groups and take 2 of those groups”
2
3
* 15 sixteenths = 2 *
1
3
∗ 15 sixteenths = 2 * 5 sixteenths = 10 sixteenths
OR
2
3
* 15
1
16
= 2 *
1
3
∗ 15
1
16
= 2 * 5
1
16
= 10
1
16
51. Moment of Reflection
Think about how the fractions
2
3
and
15
16
are
interpreted in the problem you just completed.
Joann has
15
16
of a whole cup of sugar. She wants
to give
2
3
of what she has to her friend Ann so
Ann can make some cookies. How much sugar
will Ann receive?
55. Review
3
4
*
8
19
Multiplier Multiplicative Unit
Interpret fraction as an operator
Partition multiplicative unit into
4 equal groups and “use” 3 of
those groups.
Interpret fraction as a scalar
multiple.
8 nineteenths or 8 copies of
1
19
3
4
*8 nineteenths = 3 *
1
4
∗ 8 nineteenths = 6 nineteenths =
6
19
57. Going Further
• Jamie has
4
5
of a whole foot of string and wants to
create pieces of string that are
1
3
of a foot long.
How many pieces of string that are
1
3
of a foot
long can she create? (Be sure to include partial
pieces.)
• Jamie has 12 feet of string and wants to create
pieces that are 5 feet long. How many of those
size pieces can she create?
58.
59.
60. Problem 11
You have 12 peanut butter and jelly sandwiches
and each student will eat 0.6 of a peanut butter
and jelly sandwich. How many students can you
feed?
61. Connecting to Problem 3
Joann has 5 cups of sugar. She wants to give
2
3
of
what she has to her friend Ann so Ann can make
some cookies. How much of a whole pound of
sugar will Anne receive?
62. Going Further
Joann has
4
5
of a pound of sugar. She wants to
give
2
3
of what she has to her friend Ann so Ann
can make some cookies. How much sugar will
Ann receive?
63. Going Even Further
• Each batch of cookies requires 0.4 of a cup of
sugar. How much sugar will be needed to
make 5 batches of cookies?
• Joann has 0.8 of a cup of sugar. She wants to
give 0.2 of what she has to her friend Ann so
Ann can make some cookies. How much of a
whole cup of sugar will Ann receive?
